Research

Research with Students

Since finishing grad school and beginning work as a professor, most of my research has been done with students. The majority of my student projects are in areas somehow adjacent to functional analysis, though with occasional exceptions.

Here are the senior theses that I've supervised:

  • 2020-21 Jonathan Meckel "Numerical Range and Connvergence of Truncated Composition Operators"

  • 2018-19 Rose Schweizer "Diophantine Equations and Factorization in Quadratic Integer Rings"

  • 2016-17 Daniel Halmrast "Spectral Decomposition of Quantum-Mechanical Operators"

  • 2014-15 Joshua Mirth "Functional Analysis and the Dirichlet Problem"

Most of the above theses began as summer projects. Here are other summer projects:

  • 2014 Noah Diekemper "Combinatorial and Computational Aspects of Generalizations of Free Probability"

  • 2020 C. J. DeStefani "Statistical Properties of Numerical Ranges of 2x2 Random Matrices"

  • 2021 Jack Graham "The Discrete Brachistochrone"

Other Research

My dissertation was on dilations of completely positive semigroups. Since its origin in the 1970's, this area of research has been driven by two major motivations: First, in physics, the dynamics of an open quantum system (an important object of study in quantum thermodynamics and quantum optics), subject to certain simplifying assumptions, is described by a completely positive semigroup, and the question of dilation has to do with the existence of a closed system which contains the given open system. Second, in pure mathematics, dilation of a completely positive semigroup is understood as construction of a noncommutative Markov process, which is of interest both for its own sake and for applications to other topics such as noncommutative partial differential equations.

I focused particularly on the dilation theory of Jean-Luc Sauvageot, which is distinguished by (1) achieving a unital dilation, in contrast to the corner dilations produced by other techniques, and (2) building the dilation out of something like a free product, which is the coproduct in the category of unital C*-algebras; since commutative dilations are built out of the tensor product, which is the coproduct in the category of commutative unital C*-algebras, this is an attractive parallel. However, Sauvageot's approach to dilation theory faces its own challenges, and its connections with other approaches remain unclear.


Isaac Newton